Theorem bj-sbimedh 15427

Description: A strengthening of sbiedh 1801 (same proof). (Contributed by BJ, 16-Dec-2019.)

Hypotheses

Ref	Expression
bj-sbimedh.1	|- ( ph -> A. x ph )
bj-sbimedh.2	|- ( ph -> ( ch -> A. x ch ) )
bj-sbimedh.3	|- ( ph -> ( x = y -> ( ps -> ch ) ) )

Assertion

Ref	Expression
bj-sbimedh	|- ( ph -> ( [ y / x ] ps -> ch ) )

Proof of Theorem bj-sbimedh

Step	Hyp	Ref	Expression
1		sb1 1780	. . 3 |- ( [ y / x ] ps -> E. x ( x = y /\ ps ) )
2		bj-sbimedh.1	. . . 4 |- ( ph -> A. x ph )
3		bj-sbimedh.3	. . . . 5 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
4	3	impd 254	. . . 4 |- ( ph -> ( ( x = y /\ ps ) -> ch ) )
5	2, 4	eximdh 1625	. . 3 |- ( ph -> ( E. x ( x = y /\ ps ) -> E. x ch ) )
6	1, 5	syl5 32	. 2 |- ( ph -> ( [ y / x ] ps -> E. x ch ) )
7		bj-sbimedh.2	. . 3 |- ( ph -> ( ch -> A. x ch ) )
8	2, 7	19.9hd 1676	. 2 |- ( ph -> ( E. x ch -> ch ) )
9	6, 8	syld 45	1 |- ( ph -> ( [ y / x ] ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1506   [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  bj-sbimeh  15428